2130_Chapter7_Notes

# Lets look at an example x 1 1 x 1 3 find the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: v2 = 0 1 −2 2 −4 1 −2 2 −4 0 v1 = v2 0 (ﬁrst equation) =⇒ v1 = 2v2 =⇒ v = Check #2: (A − 2I)v = 3−2 −2 2 −2−2 2c 2 =c , c = 0. c 1 0 2 . = 0 1 9 / 12 5 Exa 2 (slide 3) λ2 = −1 =⇒ A − λ2 I = A + I = Check #1: det(A + I) = 3+1 −2 2 −2+1 4 −2 2 −1 4 −2 = −4 + 4 = 0. 2 −1 (A + I)v = 0 =⇒ 4 −2 2 −1 =⇒ 2v1 − v2 = 0 (second equation) =⇒ v2 = 2v1 =⇒ v = Check #2: (A + I)v = = 4 −2 2 −1 v1 0 = v2 0 c 1 =c , c = 0. 2c 2 1 0 = . 2 0 10 / 12 Exa 2 (slide 4) λ1 = 2, v(1) = λ2 = −1, v(2) = Fundamental matrix: Ψ(t) = 2 1 =⇒ x(1) = 1 2 =⇒ x(2) = 2 2t e 1 1 −t e 2 2e2t e−t . e2t 2e−t Next, compute the special fundamental matrix Φ(t) at t0 = 0. 11 / 12 6 Exa 2 (slide 4) Fundamental matrix: Ψ(t) = 2e2t e−t . e2t 2e−t Special fundamental matrix: Φ(t) = Ψ(t)Ψ−1 (0). Ψ−1 (0) 21 = 12 =⇒ Φ(t) = = Check: Φ(0) = −1 = 1 2 −1 2 3 −1 1 2e2t e−t 3 e2t 2e−t 2 −1 −1 2 1 4e2t − e−t 2(e−t − e2t ) 3 2(e2t − e−t ) 4e−t − e2t 1 4−1 2(1 −...
View Full Document

Ask a homework question - tutors are online